Spreading Factor Allocation Strategy for LoRa Networks under Imperfect Orthogonality

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Spreading Factor Allocation Strategy for LoRa

Networks under Imperfect Orthogonality

Licia Amichi, Megumi Kaneko, Nancy Rachkidy, Alexandre Guitton

To cite this version:

Licia Amichi, Megumi Kaneko, Nancy Rachkidy, Alexandre Guitton. Spreading Factor Allocation

Strategy for LoRa Networks under Imperfect Orthogonality. IEEE International Conference on

Com-munications, May 2019, Shanghai, China. �hal-02022622�

Spreading Factor Allocation Strategy for LoRa

Networks under Imperfect Orthogonality

Licia Amichi

, Megumi Kaneko

, Nancy El Rachkidy

, and Alexandre Guitton

† _{Universit´e Clermont Auvergne, CNRS, LIMOS, F-63000 Clermont-Ferrand, France}

Email: licia.amichi@etu.upmc.fr, megkaneko@nii.ac.jp, {nancy.el rachkidy, alexandre.guitton}@uca.fr

Abstract—Low-Power Wide-Area Network (LPWAN) based on LoRa physical layer is envisioned as one of the most promising technologies to support future Internet of Things (IoT) systems. LoRa provides flexible adaptations of coverage and data rates by allocating different Spreading Factors (SFs) to end-devices. Although most works so far had considered perfect orthogonality among SFs, the harmful effects of inter-SF interferences have been demonstrated recently. Therefore in this work, we consider the problem of SF allocation optimization under co-SF and inter-SF interferences, for uplink transmissions from end-devices to the gateway. To provide fairness, we formulate the problem as maximizing the minimum achievable average rate in LoRa, and propose a SF allocation algorithm based on matching theory. Numerical results show that our proposed algorithm enables to jointly enhance the minimal user rates, network throughput and fairness, compared to baseline SF allocation methods.

Index Terms—LoRa, Spreading Factor, Imperfect Orthogonal-ity, Resource Allocation Optimization, Matching Theory

I. INTRODUCTION

Tens of billions of Internet of Thing (IoT) devices are expected by 2020, for enabling groundbreaking applications such as smart cities, intelligent transportation systems or harsh environment monitoring. Many of these IoT Wireless Sensor Networks (WSNs) will strongly rely on Low-Power Wide-Area Network (LPWAN) technologies such as Long Range (LoRa) [1, 2]. LoRa is one of the most prominent physical layer technologies for LPWAN, as it operates in the ISM unlicensed bands and enables flexible adaptations of transmission rates and coverages under low energy con-sumption [3]. LoRa is based on Chirp Spread Spectrum (CSS) modulation, where each chirp encodes 2msymbol values, for Spreading Factor (SF) m. On top of LoRa physical layer, LoRaWAN defines the MAC layer protocol standardized by LoRa Alliance [4]. The LoRaWAN architecture is a star-like topology, where end-devices communicate with gateways over several channels based on ALOHA mechanism, with duty cycle limitations [1]. In LoRaWAN, six SFs ranging from m = 7 to 12 are used, where smaller SFs provide higher data rates but reduced ranges, while larger SFs allow longer ranges but lower rates [2].

One of the main issues in LoRa and LoRaWAN is the throughput limitation: the physical bitrate varies between 300 and 50000 bps [4]. In addition, collisions are very harmful to the system performance as the LoRa gateway is unable to

Fig. 1. LoRa system, with end-devices transmitting simultaneously

correctly decode simultaneous signals sent by devices using the same SF and the same channel. Such interferences will be referred to as co-SF interferences. Although SFs were widely believed to be orthogonal among themselves, some recent studies have shown that this is not the case by exper-imentally evaluating the effects of inter-SF interferences [5, 6]. Thus, authors in [7] have analyzed the achievable uplink LoRa throughput under imperfect SF orthogonality, and have demonstrated the harmful impact of both co-SF and inter-SF interferences on the overall throughput.

In order to improve the LoRa system performance, a number of works have proposed resource optimization methods [8– 10]. However, most papers so far have assumed a perfect orthogonality among SFs. In particular, the authors in [8] design a channel and power allocation algorithm, that max-imizes the minimal rate. However, the allocated channels are perfectly orthogonal, and no SF allocation nor SF-dependent rates were not considered, despite the strong dependency of the rate to SFs. In addition, the solution of [8] requires instan-taneous Channel State Information (CSI) feedback, which is not adapted to LoRa networks due to their energy consumption limitations [4]. In [9], a heuristic SF-allocation is proposed, where users with similar path losses are simply assigned to the same channel, then to each SF according to their distance to the gateway. Although the issue of inter-SF interferences was highlighted, it was ignored in their proposed solution. In [10], a method for decoding superposed LoRa signals using

the same SF was proposed.

Therefore, in this work, we consider the issue of SF alloca-tion optimizaalloca-tion under both co-SF and inter-SF interferences. In particular, we focus on the problem of maximizing the minimum achievable short-term average rate in the uplink, whereby short-term average rate is defined as the average rate over random channel fadings, but given a fixed position of end-devices. This metric is especially suited for LoRa networks, since the end-devices will likely be fixed for a certain period of time (at least for a few seconds) in many applications, and their positions known at the gateway, as in the conventional signal-strength-based SF allocation method [4]. Our focus here is the max-min fairness in terms of throughput, without taking into consideration the energy of end-devices. As LoRaWAN enforces a limited duty-cycle, the energy consumption of each end-devices remains limited. Firstly, we formulate our optimization problem by modeling the achievable uplink short-term average rate under co-SF and inter-SF interferences. Next, given the mathematical intractability of the optimiza-tion problem, we propose an SF-allocaoptimiza-tion algorithm based on matching theory. We show its stability and convergence properties, and analyze its computational complexity. Finally, numerical results demonstrate that, compared to baseline SF-allocation schemes, our proposed method not only provides larger minimum rates, but also jointly improves the network throughput and fairness level.

II. SYSTEMMODEL

We consider an uplink LoRa system with a single gateway located at the center of a cell of radius R km and N end-devices uniformly distributed and simultaneously active as depicted in Figure 1. End-devices transmit on the same channel c of bandwidth BW , with a power Pmax and a duty cycle of

100%. We denote the set of SFs by M = {7, 8, . . . , 12}, where M = |M| = 6. Each SFm, m ∈ M, has a data bit-rate

equal to Rm [3], Rm= m × CR 2m BW , (1)

where the Coding Rate is given in [3] as CR = _4+n4 with n ∈ {1, 2, 3, 4}.

The uplink instantaneous Signal-to-Noise Ratio (SNR), γnm, for end-device n at SFm is given by [7],

γnm=

Pmax|hn|2A(fc)

rα nσc2

, (2)

where hn is the channel gain between n and the gateway,

A(fc) = (fc2 × 10−2.8)−1 is the deterministic loss,

fc is the carrier frequency, rn the distance from

end-device n to the gateway, α the path loss exponent and σ_c2= −174 + NF + 10log(BW ) [dBm] is the Additive White Gaussian Noise (AWGN) where NF is the receiver Noise Figure. Assuming Rayleigh fading channels, the SNR γnm

is modeled as an exponential random variable with mean γ_nm=A(fc)Pmax rα nσ2c . SF m Bit-rate [kb/s] Receiver sensitiv-ity [3] [dBm] Reception thresh. θrxm[dB] InterSF thresh. [11] ˜ θm[dB] Distance ranges 7 5.47 -123 -6 -7.5 0-l7 8 3.13 -126 -9 -9 l7-l8 9 1.76 -129 -12 -13.5 l8-l9 10 0.98 -132 -15 -15 l9-l10 11 0.54 -134.5 -17.5 -18 l10-l11 12 0.29 -137 -20 -22.5 l11-l12 TABLE I LORACHARACTERISTICS ATBW =125KHZ[7]

The area covered by each SF is given by the distance ranges in Table I with [7], lm= e 1 α×ln A(fc)LBm
, (3)

where LBmis the link budget of the SFmdefined as LBm=

Pmax− θrxm, given the receiver sensitivity of each SFmθrxm

in Table I. Hence, larger SFs result in larger communication ranges, with l12= R.

If there is only one device n assigned to SFm, this device is

only subject to inter-SF interferences caused by devices using a different SF. Hence the inter-SF Signal-to-Interference-plus-Noise-Ratio (SINR) of end-device n can be expressed as

SINRinter_nm = _P γnm i∈N−n P j∈M−m sijγij+ 1 , (4)

where N−n= N \{n}, M−m= M\{m} and sij is a binary

variable used to indicate whether n uses m, sij=



1, if i uses j 0, otherwise.

When there is more than one device assigned to a SF, these devices are subject to both inter-SF and co-SF interferences. However, as shown in [5–7], co-SF interferences are largely dominant over inter-SF interferences1. Therefore, the co-SF SINR of end-device n on SFmis written as,

SINRco_nm= _P γnm

i∈N−n

simγim+ 1

. (5)

In conformity to LoRaWAN standards, instantaneous CSI feedback is not assumed, unlike [8]. Hence, the SF allocation is performed every period of time, during which the long-term fading instance, i.e., path loss distance, can be assumed to be fixed. Therefore, the achievable uplink short-term average rate for device n at SFmis given similarly to [7] by,

τnm= Rm× Pcap(n,m), (6)

where Rm is the data bit-rate of the SFm and P (n,m) cap the

probability of successful transmission, which is analyzed in the following section.

1_{LoRa chirps were especially designed to yield limited inter-SF}

III. PROBLEMFORMULATION

In this section, we formulate the considered optimization problem. Since fairness will be crucial in LoRa systems, we strive to maximize the minimal uplink short-term average rate over devices and SFs, under co-SF and inter-SF interferences. We first derive the expression of Pcap(n,m). Assuming N > 1,

there are two cases:

1) One device n at SFm: device n is only subject to

inter-SF interferences. The transmission can be successfully decoded if the node satisfies the inter-SF as well as the reception condition. In this case inter-SF interferences are more critical than the reception condition since there will always be inter-SF interferences for N > 1. Hence the probability of successful transmission can be written as,

Pcap(n,m)_iSF= P



SINRinternm ≥ ˜θm



, (7)

where SINRinternm is given in (4) and ˜θm is the inter-SF

interference capture threshold for SFm, defined in Table I.

Using the random instantaneous SNR variables γnm for all

(n, m) and marginalizing over them, we show in the Appendix with similar calculations as in [7] that (7) can be written as,

Pcap(n,m)iSF = e − θmσ˜ 2c rαn A(fc)Pmax Y (i,j)∈ N−n×M−m 1 ˜ θmsij(rn_r i) α + 1. (8)

2) More than one device at SFm: in this case, co-SF

in-terferences largely dominate the inter-SF inin-terferences as well as reception condition as explained in Section II. Therefore, the success probability is expressed as in [12],

Pcap(n,m)_coSF= P (SINR co

nm≥ θco) , (9)

where SINRco_nm is given in (5) and θco is the co-SF capture

threshold which is equal to 6dB for all SFm [3, 12]. With

similar calculations as in the Appendix, we obtain

Pcap(n,m)coSF= e − θcoσc r2 αn A(fc)Pmax Y i∈N−n 1 θcosim×  rn ri α + 1 . ₍₁₀₎

Given the above analysis, we can now formulate our objective function as follows (for N > 1),

max snm min (n,m)∈ N ×M s.t.snm6=0 snmRmPcap(n,m), (11)

where the minimization is over the snmthat are non-zero, and

Pcap(n,m)= I   X k∈N skm≥ 2   P (n,m) cap_coSF+ I   X k∈N skm= 1   P (n,m) cap_iSF, (12)

where I(C) is the indicator function, i.e., it equals 1 if the condition C is verified and 0 otherwise.

Finally, the overall optimization problem becomes

(P ) max snm min (n,m)∈ N ×M s.t.snm6=0 snmRm " I   X k∈N skm≥ 2   e − θcoσc r2nα A(fc)Pmax × Y i∈N−n 1 θcosim(r_rn i) α_{+ 1}+ I   X k∈N skm= 1   e − θmσ˜ c r2 αn A(fc)Pmax × Y (i,j)∈ N−n×M−m 1 ˜ θmsij  rn ri α + 1 # (13) s.t. C1: snm∈ {0, 1} ∀(n, m) ∈ N × M (13a) C2: X m∈M snm≤ 1, ∀n ∈ N (13b) C3: X n∈N snm≤ Nmax(m), ∀m ∈ M (13c) C4: if N > M, 1 ≤ X n∈M snm, ∀m (13d)

This optimization problem is constrained by the binary al-location variable snm ∈ {0, 1} (13a), the fact that an

end-device n is assigned to at most one SF (13b) and that the maximal number of devices at SFm is upper bounded by

Nmax(m) (13c)2, if there are enough devices (N > M ),

no SFs should remain unused (13d). (P ) is hence an integer programming problem, given the binary variables snm, with a

non-linear objective function. This is an NP-hard problem [13], hence it is difficult to obtain its optimal solution directly. Therefore, we propose an optimized SF allocation method, using tools from matching theory.

IV. PROPOSEDMETHOD

Matching theory is a promising tool for resource allocation in wireless networks [14]. According to this theory, our considered allocation problem (P ) can be classified as a many-to-one matching problem with conventional externalities and peer effects. There are two sets of players, the set of SFs and the set of end-devices, where each element seeks to be matched with players of the opposing set. A device will prefer to be matched to the SF offering the highest utility, while each SF prefers to be matched with the group of devices with the highest utility. The difficulty of our problem is that there is an interdependence between nodes’ preferences, i.e., whenever a device is matched to an SF, the preferences of the other devices may change due to co-SF interferences. In addition to these conventional externalities (preference interdependence) and unlike the problem in [8] where only orthogonal channels (not SFs) were considered, our problem exhibits peer effects that are caused by inter-SF interferences. That is, the preferences of a device depend not only on the identity of the SF and the number of devices assigned to it, but also on the assignment of devices to other SFs (since they cause inter-SF interferences). Hence, to solve (P ), we propose a many-to-one matching

2_{Setting N}

max(m) enables to control the harmful effects of co-SF

inter-ferences, and reduces the computational complexity of the proposed method, as shown in Sections IV-B and V-C.

algorithm between the set of SFs M and the set of end-devices N . First, we define the basic concepts of matching theory. Matching pair: a couple (n, m) assigned to each other. Quotas of a player: the maximum number of players with which it can be matched

• Each end-device has a quota of 1 (13b),

• Each SFmhas a quota of Nmax(m) devices (13c).

Utility of a node: defined for our problem as its short-term average rate. If it is the only device at SFm,

Un= Rme − θmσ˜ c r2nα A(fc)Pmax Y (i,j)∈ N−n×M−m 1 ˜ θmsij  rn ri α + 1 . (14)

If it shares the SFm with other devices,

Un= Rme − θcoσ 2 c rαn A(fc)Pmax Y i∈N−n 1 θcosim(rn_r i) α + 1. (15)

Utility of an SF: defined for our problem as the minimum short-term average rate among the devices assigned to it. If SFm is matched to one device only:

Um= min n∈N∗ m Rme − θmσ˜ 2 c rαn A(fc)Pmax Y (i,j)∈ N−n×M−m 1 ˜ θmsij  rn ri α + 1 , (16) where N∗m is the set of devices assigned to the SFm,

otherwise Um is given as Um= min n∈N∗ m Rme − θcoσ 2 c rnα A(fc)Pmax Y i∈N−n 1 θcosim(rn_r i) α + 1. (17)

Preference relation: a player q prefers a player p1 over the

player p2, if the utility of q is higher when it is matched to

p1 than when it is matched to p2.

Blocking pair: a matching pair (n, m) is a blocking pair when Unor Umis higher when n uses m, than when they use their

current matches, without lowering the utilities of any other device nor SF. In this case, n will leave its current match to be matched to m.

Two-sided stable matching: a matching solution where there is no blocking pair.

A. Proposed algorithm

In this subsection, we describe the steps of the proposed matching-based algorithm which exploits matching techniques as in [8, 14], tailored to our specific problem. First, the scheduler located at the gateway performs an initial matching between the set of SFs M and the set of end-devices N by the Initial Matching in Algorithm 1. Next, it swaps the matching pairs obtained in the previous step until reaching a two-sided exchange stable matching by Matching Refinement in Algorithm 2. Details of these steps are given below.

Let LU denote the set of end-devices that are not allocated

to any SF, req_m the requests received by the SFm, and Am

the set of devices assigned to the SFm. We suppose that the

gateway knows the distance of all end-devices to it.

Initialization: the gateway starts by initializing the pref-erence lists of end-devices and SFs. Each device n with a distance rn to the gateway, can only use SFs including it in

their coverage area (rn≤ lm), therefore,

Lp,n= {m ∈ M, st rn≤ lm}, (18)

Lp,n is arranged according to the increasing order of the

distance threshold of the SFs (lm, m ∈ M ), i.e, an SF with

higher achievable rate is preferred. On the other hand, an SFm

only considers devices having a distance to the gateway lower than lm,

Lp,m= {n ∈ N , st rn≤ lm}. (19)

Lp,m is ordered such that, a user n1∈ Lp,m is ranked before

another user n2 ∈ Lp,m if n1 is located in the ring of SFm

(n1∈ (lm−1, lm]) but not n2 (n2∈ (l/ m−1, lm]), or both are

in the ring of SFm but n1 is closer to the gateway than n2

(|lm−1− rn1| < |lm−1− rn2|).

Unmatched end-devices are added to LU.

Initial Matching: for each device n in the unmatched list LU, if Lp,n 6= ∅, n requests its first preferred SF and

removes it from Lp,n, otherwise the device is removed from

LU since all SFs it can use have already reached their quota.

Then, each SFm either accepts all current requests if its

quota allows it, or it accepts the requests of its most preferred devices that fulfill its quota, if not. This process is repeated until LU becomes empty. Note that, setting this quota of

Nmax(m) devices for each SFm also avoids large contention

delays.

Matching Refinement: for each matching pair (n, m) the algorithm calculates Um using (16) if it is only assigned to

device n and (17) in the other case. The utility of device n is calculated by (14) if it is the only one at SFm, and with (15)

otherwise. Firstly, if there is an SFlthat is not assigned to any

device that allows increasing Un, the device leaves SFmto be

matched with SFl. Then, the algorithm calculates the utilities

of every pair (k, l), and makes a swap between (n, m) and (k, l) and determines their new utilities. If (k, m) or (n, l) is a blocking pair the algorithm validates the swap. This swapping step is repeated until reaching a two-sided stable matching. B. Algorithm analysis

We now prove the stability and convergence of the proposed algorithm, and analyze its computational complexity.

Proposition 1. Stability: When the proposed algorithm termi-nates, it finds a two-sided exchange stable matching.

Proof. Let’s assume that the proposed algorithm terminates and the final matching is not two-sided exchange stable. Then, the matching contains at least one more blocking pair (k, m) or (n, l) where the utility of at least one player among {n, m, k, l}, can be improved without lowering others’ utility. Accordingly, the proposed algorithm did not terminate and will continue, thereby the matching is not final, which contradicts the initial assumption.

Algorithm 1 Initial Matching

while LU is not emptydo

for i ∈ LU do

if Lp,i== ∅ then

LU← LU\{i};

else

a ← firstPrefered(Lp,i); // Favorite SF

Lp,i← Lp,i\{a};

reqa← reqa∪ {i};

for j ∈ M do

if size(req_j) > 0 and size(Aj) < NMax(j) then

if (size(reqj) + size(Aj)) ≤ NMax(j) then

Accept all the requests and add the devices toAj;

else

Accept the requests of the NMax(j)-size(Aj)

most preferred devices, and add them toAj;

Algorithm 2 Matching Refinement

while change == true do change = false; for j ∈ M do CalculateUjeq (16) or eq (17); for i ∈ Ajdo CalculateUieq (14) or eq (15); for l ∈ M−j do if size(Al) == 0 then Swap (i, j),(∅, l)
;

Calculate the new utility of i, U0i eq (14) or

eq (15); if U0i≥ Ui then

Validate the Swap; change ← true; else CalculateUleq (16) or eq (17); for k ∈ Aldo Calculate Uk eq (14) or eq (15); Swap (i, j),(k, l)
;

if (i,l) or (k,j) is a blocking pair then Validate the Swap;

change ← true;

Proposition 2. Convergence: After a finite number of swap operations, the algorithm eventually converges to a two-sided exchange stable matching.

Proof. A swap operation occurs if it improves the utility of at least one player without decreasing the others’, hence the utilities can only rise. Additionally, the maximal throughput that can be achieved on an SFmis upper-bounded by the data

bit-rate Rm, meaning that each SFmand the devices assigned

to it have utilities upper bounded by Rm.

The number of potential swap operations is finite: device assigned to SFlcan make at most Nmax(l) × P

j∈M−l

Nmax(j)

swap operations. The total number of swap operations is thus upper-bounded by P

l∈M

Nmax(l) × P j∈M−l

Nmax(j).

Proposition 3. Complexity: The running time of our proposed algorithm is upper-bounded byO N M + Q2_M2
_.

Proof. Initial matching complexity: in the worst case, all the devices have the same preference list, and they are located in the area covered by all the SFs. At round1the gateway receives

N requests, at round2 it receives N − Nmax(m1) requests, at

roundi it receives N − M −1

P

k=1

Nmax(mi) requests. Therefore,

the total number of requests equals N M −

M −1

P

i=1

(M − i) × Nmax(mi). The complexity of the initial matching is upper

bounded by: O N M ).

Matching refinement complexity: in each iteration, for each SFm, the algorithm considers at most Nmax(m) devices

and examines P

l∈M−m

Nmax(l) swap operations for each of

these devices. Therefore, the number of swap operations that are examined in one iteration is upper bounded by

P

m∈M

Nmax(m)× P l∈M−m

Nmax(l). Let Q = max

m∈M{Nmax(m)},

thus the computational complexity of the matching refinement is upper bounded by O Q2_{M (M − 1)).}

In summary, the computational complexity of our algorithm is upper bounded by O N M + Q2M2
.

Note that this complexity is not excessive as our algorithm is run at the gateway which is not computationally-limited.

V. NUMERICALRESULTS

A. Simulation Settings

We consider a cell of radius R=1km [12], with a varying number of devices N from N=2 to 40. Note that all devices transmit with a duty cycle of 100%. Hence, with a duty cycle of 1% as preconized in LoRaWAN [1], the actual number of end-devices would be 100-fold3, i.e., up to 4000. All end-devices transmit in the channel of carrier frequency fc=868[MHz] with a bandwidth BW =125[kHz] and power

Pmax = 14 [dBm]. We consider a lossy urban environment,

therefore we take a path loss exponent equal to 4 as in [7, 12]. The computer simulation environment has been implemented in Matlab4_.

B. Baseline schemes

We consider two baseline schemes for performance com-parison the random SF allocation [7], and the distance-SF allocation algorithms [11], with a maximal number of simul-taneously transmitting devices equal to A (specified later): Random SF: the gateway chooses randomly A devices among N and assigns a random SF to each of these devices among the possible SFs.

Distance SF: the gateway chooses randomly A devices among N . Then, the SF for each of these devices is determined by

3_{The evaluations are made for 100% duty cycle as this is the most}

challenging case. Hence, much better performance can be expected in the case of 1% duty cycle.

4_{All results are based on our mathematical model detailed in the Appendix}

NM ax SF7 SF8 SF9 SF10 SF11 SF12

1 4.82 1.51 1.06 4.7e-1 2.7e-1 1.9e-1 2 7.7e-2 1.1e-7 9.3e-14 7.8e-25 6.7e-46 3.7e-78 3 2.7e-3 8.2e-9 2e-15 8.2e-27 3.1e-49 1.3e-84 4 9.9e-5 5.8e-10 9.0e-17 4.3e-29 1.2e-49 1.1e-86 5 1.8e-6 5.2e-11 6.5e-18 1.3e-30 1.0e-53 3.7e-93

TABLE II

MINIMAL THROUGHPUT FOR EACHSFm[KBIT/S]

Table I based on their distance rn: device n uses SFm if

rn∈ (lm−1, lm].

C. Choice of Nmax given a target minimum throughput

To determine the quota of each SF, we fix a target minimal throughput equal to 1 [bit/s]. We have run preliminary simu-lations over 100000 frames. Table II represents the minimal short-term average rate achieved on each SF, for different values of Nmax. We can observe that to guarantee the target

minimal throughput of 1 [bit/s], we can have at most three devices assigned to SF7 but only one device for the other

SFs. In the sequel, the proposed method is compared to the baseline SF-allocation schemes. For fair comparison, the number of simultaneously transmitting devices A is equal to

P

m∈M

Nmax(m) = 8 in each allocation period.

D. Simulation results 10 20 30 40 Number of nodes 0 0.1 0.2 0.3 0.4 Minimal Throughput[kbits/s] Conv. Random Conv. Distance Prop. Initial Prop. Allocation

Fig. 2. Minimal short-term average rates, proposed and baseline algorithms

Figure 2 depicts the performance comparison of our SF-allocation algorithm, and the baseline schemes in terms of minimal short-term average rates, against a varying number of end-devices. First, we can observe that the proposed allo-cation outperforms both the conventional random and distance allocations for all values of N . While baseline schemes lead to an early drop of minimal rate (almost null for N > 6), the proposed algorithms still provide a good minimal throughput for a much higher number of devices. We can also notice that our Matching Refinement (Prop. Allocation) provides a better minimal throughput than the Initial Matching (Prop. Initial).

Next, Figure 3 shows the impact of maximizing the minimal

5 10 15 20 25 30 35 40 Number of nodes 0 0.2 0.4 0.6 0.8 1 1.2 1.4

Average Throughput [kbits/s]

Conv. Random Conv. Distance Prop. Initial Prop. Allocation

Fig. 3. Average network throughput, proposed and baseline algorithms

short-term average rates on the average network throughput over all end-devices. It can be clearly seen that the highest average network throughput is achieved by our proposed SF-allocation method. Also, it can be noted that our Matching Refinement provides a significant improvement compared to the Initial Matching.

Finally, we evaluate the fairness levels of the different

algo-5 10 15 20 25 30 35 40 Number of nodes 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Jain's fairness Conv. Random Conv. Distance Prop. Initial Prop. Allocation

Fig. 4. Jain’s fairness metric, proposed and baseline algorithms

rithms by the Jain’s fairness index, given by J =

P n∈N Un !2 N ×P n∈N U2 n . Figure 4 shows that by the considered max-min strategy and matching-based methodology, the proposed algorithms significantly enhance the system fairness level compared to baseline methods.

Overall, the proposed SF-allocation method provides remark-able performance improvements, jointly in terms of minimal achievable rates, network throughput and fairness, with a

limited computational complexity suitable for LoRa gateways.

VI. CONCLUSION

We have considered the problem of fairness-aware SF-allocation in a LoRa-based network composed of IoT end-devices. Unlike most previous works, we have integrated both the effects of co-SF and inter-SF interferences in the expression of the achievable uplink short-term average rate of each end-device. We have then formulated our SF allocation problem as a max-min optimization of the uplink short-term average rate of end-devices. To solve this NP-hard optimiza-tion problem, we have proposed an SF-allocaoptimiza-tion algorithm based on matching theory, and proved its stability, convergence properties and its reasonable complexity. Numerical results show that our SF-allocation strategy not only outperforms the conventional algorithms in terms of minimal rate, but also jointly in terms of network throughput and fairness. For future work, this problem will be extended by jointly controlling the transmission power and SF allocation and by designing distributed algorithms. Energy efficiency aspects will be also investigated.

REFERENCES

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APPENDIX

We now prove the expression (8). Using the random in-stantaneous SNR variables γnm for all (n, m), (7) can be

developed as P       γnm P (i,j)∈ N−n×M−m sijγij+ 1 ≥ ˜θm      γ11, . . . γn−1m−1, γn+1m+1, . . . γN M       × P (γ11, . . . γn−1m−1, γn+1m+1, . . . γN M) . (20)

The variables γnm are modeled as exponential random

vari-ables with mean γ_nm. Therefore, by marginalizing over these variables we obtain Z γ11 · · · Z γn−1m−1 Z γn+1m+1 · · · Z γNM P        γnm ≥ ˜θm        X (i,j)∈ N−n×M−m sij γij + 1               | {z } A × Pγ11, . . . γn−1m−1, γn+1m+1, . . . γNMdγ11 . . . dγNM . (21)

First, we calculate A where pγ(γnm) = γ_nm1 e −γnm γnm A = ∞ Z ˜ θm   P (i,j)∈N−n×M−m sijγij+1   pγ(γnm) dγnm = e− ˜ θm γnm Y (i,j)∈N−n×M−m 1 ˜ θmsijγij γ_nm + 1 . (22)

By replacing A in Pcap(n,m)_iSF, we have

Pcap(n,m)iSF= e −θm˜ γnm Y (i,j)∈ N−n×M−m Z γij e− ˜ θmsij γij γnm pγ(γij) dγij = e− ˜ θm γnm Y (i,j)∈ N−n×M−m 1 ˜ θmsijγij γ_nm + 1 . (23) Finally, we obtain Pcap(n,m)_iSF= e − θmσ˜ c r2nα A(fc)Pmax Y (i,j)∈ N_−n×M_−m 1 ˜ θmsij(r_rn i) α + 1. (24)